LMFDB - Embedded newform 6800.2.a.bz.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6800,2,Mod(1,6800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,0,0,0,-1,0,6,0,-4,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.60789$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60789 q^{3} +1.33298 q^{7} +3.80107 q^{9} -1.09485 q^{11} +3.17083 q^{13} +1.00000 q^{17} -2.75613 q^{19} -3.47626 q^{21} -3.57111 q^{23} -2.08911 q^{27} -0.180058 q^{29} -0.816039 q^{31} +2.85524 q^{33} +8.44817 q^{37} -8.26917 q^{39} -7.97191 q^{41} -6.54798 q^{43} -0.576074 q^{47} -5.22317 q^{49} -2.60789 q^{51} +7.84602 q^{53} +7.18768 q^{57} +9.76375 q^{59} -5.21577 q^{61} +5.06675 q^{63} +2.64963 q^{67} +9.31305 q^{69} -13.4114 q^{71} -12.3299 q^{73} -1.45941 q^{77} -3.74745 q^{79} -5.95506 q^{81} +4.66596 q^{83} +0.469570 q^{87} +3.00946 q^{89} +4.22665 q^{91} +2.12814 q^{93} -12.2681 q^{97} -4.16160 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - q^{7} + 6 q^{9} - 4 q^{11} - 3 q^{13} + 5 q^{17} - 6 q^{19} - 5 q^{21} - 4 q^{23} + 5 q^{27} + 2 q^{29} - 21 q^{31} + 12 q^{33} - 2 q^{37} - 23 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} + 10 q^{49}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q - q^3 - q^7 + 6 * q^9 - 4 * q^11 - 3 * q^13 + 5 * q^17 - 6 * q^19 - 5 * q^21 - 4 * q^23 + 5 * q^27 + 2 * q^29 - 21 * q^31 + 12 * q^33 - 2 * q^37 - 23 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 + 10 * q^49 - q^51 + 21 * q^53 - 20 * q^57 - 12 * q^59 - 2 * q^61 - 8 * q^63 + 12 * q^67 - 21 * q^71 + 22 * q^73 - 32 * q^77 - 41 * q^79 - 35 * q^81 + 8 * q^83 + 4 * q^87 - 10 * q^89 - 13 * q^91 + 9 * q^93 - 20 * q^97 + 14 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.60789	−1.50566	−0.752832	−	0.658213i	$-0.771313\pi$
$3$	−2.60789	−1.50566	−0.752832	+	0.658213i	$0.771313\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.33298	0.503819	0.251909	−	0.967751i	$-0.418941\pi$
$7$	1.33298	0.503819	0.251909	+	0.967751i	$0.418941\pi$
$8$	0	0
$9$	3.80107	1.26702
$10$	0	0
$11$	−1.09485	−0.330110	−0.165055	−	0.986284i	$-0.552780\pi$
$11$	−1.09485	−0.330110	−0.165055	+	0.986284i	$0.552780\pi$
$12$	0	0
$13$	3.17083	0.879430	0.439715	−	0.898137i	$-0.355079\pi$
$13$	3.17083	0.879430	0.439715	+	0.898137i	$0.355079\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.75613	−0.632300	−0.316150	−	0.948709i	$-0.602390\pi$
$19$	−2.75613	−0.632300	−0.316150	+	0.948709i	$0.602390\pi$
$20$	0	0
$21$	−3.47626	−0.758582
$22$	0	0
$23$	−3.57111	−0.744628	−0.372314	−	0.928107i	$-0.621436\pi$
$23$	−3.57111	−0.744628	−0.372314	+	0.928107i	$0.621436\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.08911	−0.402050
$28$	0	0
$29$	−0.180058	−0.0334359	−0.0167179	−	0.999860i	$-0.505322\pi$
$29$	−0.180058	−0.0334359	−0.0167179	+	0.999860i	$0.505322\pi$
$30$	0	0
$31$	−0.816039	−0.146565	−0.0732825	−	0.997311i	$-0.523347\pi$
$31$	−0.816039	−0.146565	−0.0732825	+	0.997311i	$0.523347\pi$
$32$	0	0
$33$	2.85524	0.497034
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.44817	1.38887	0.694435	−	0.719555i	$-0.255654\pi$
$37$	8.44817	1.38887	0.694435	+	0.719555i	$0.255654\pi$
$38$	0	0
$39$	−8.26917	−1.32413
$40$	0	0
$41$	−7.97191	−1.24500	−0.622501	−	0.782619i	$-0.713883\pi$
$41$	−7.97191	−1.24500	−0.622501	+	0.782619i	$0.713883\pi$
$42$	0	0
$43$	−6.54798	−0.998557	−0.499279	−	0.866441i	$-0.666402\pi$
$43$	−6.54798	−0.998557	−0.499279	+	0.866441i	$0.666402\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.576074	−0.0840290	−0.0420145	−	0.999117i	$-0.513378\pi$
$47$	−0.576074	−0.0840290	−0.0420145	+	0.999117i	$0.513378\pi$
$48$	0	0
$49$	−5.22317	−0.746166
$50$	0	0
$51$	−2.60789	−0.365177
$52$	0	0
$53$	7.84602	1.07773	0.538867	−	0.842391i	$-0.318853\pi$
$53$	7.84602	1.07773	0.538867	+	0.842391i	$0.318853\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.18768	0.952031
$58$	0	0
$59$	9.76375	1.27113	0.635566	−	0.772046i	$-0.280767\pi$
$59$	9.76375	1.27113	0.635566	+	0.772046i	$0.280767\pi$
$60$	0	0
$61$	−5.21577	−0.667811	−0.333906	−	0.942606i	$-0.608367\pi$
$61$	−5.21577	−0.667811	−0.333906	+	0.942606i	$0.608367\pi$
$62$	0	0
$63$	5.06675	0.638351
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.64963	0.323704	0.161852	−	0.986815i	$-0.448253\pi$
$67$	2.64963	0.323704	0.161852	+	0.986815i	$0.448253\pi$
$68$	0	0
$69$	9.31305	1.12116
$70$	0	0
$71$	−13.4114	−1.59164	−0.795820	−	0.605534i	$-0.792960\pi$
$71$	−13.4114	−1.59164	−0.795820	+	0.605534i	$0.792960\pi$
$72$	0	0
$73$	−12.3299	−1.44311	−0.721553	−	0.692359i	$-0.756571\pi$
$73$	−12.3299	−1.44311	−0.721553	+	0.692359i	$0.756571\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.45941	−0.166315
$78$	0	0
$79$	−3.74745	−0.421621	−0.210810	−	0.977527i	$-0.567610\pi$
$79$	−3.74745	−0.421621	−0.210810	+	0.977527i	$0.567610\pi$
$80$	0	0
$81$	−5.95506	−0.661673
$82$	0	0
$83$	4.66596	0.512156	0.256078	−	0.966656i	$-0.417570\pi$
$83$	4.66596	0.512156	0.256078	+	0.966656i	$0.417570\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.469570	0.0503432
$88$	0	0
$89$	3.00946	0.319002	0.159501	−	0.987198i	$-0.449012\pi$
$89$	3.00946	0.319002	0.159501	+	0.987198i	$0.449012\pi$
$90$	0	0
$91$	4.22665	0.443074
$92$	0	0
$93$	2.12814	0.220678
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.2681	−1.24564	−0.622819	−	0.782366i	$-0.714013\pi$
$97$	−12.2681	−1.24564	−0.622819	+	0.782366i	$0.714013\pi$
$98$	0	0
$99$	−4.16160	−0.418257
$100$	0	0
$101$	2.98113	0.296634	0.148317	−	0.988940i	$-0.452614\pi$
$101$	2.98113	0.296634	0.148317	+	0.988940i	$0.452614\pi$
$102$	0	0
$103$	13.8440	1.36409	0.682045	−	0.731310i	$-0.261091\pi$
$103$	13.8440	1.36409	0.682045	+	0.731310i	$0.261091\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.87103	0.760921	0.380461	−	0.924797i	$-0.375765\pi$
$107$	7.87103	0.760921	0.380461	+	0.924797i	$0.375765\pi$
$108$	0	0
$109$	13.6739	1.30972	0.654859	−	0.755751i	$-0.272728\pi$
$109$	13.6739	1.30972	0.654859	+	0.755751i	$0.272728\pi$
$110$	0	0
$111$	−22.0319	−2.09117
$112$	0	0
$113$	19.5722	1.84120	0.920600	−	0.390508i	$-0.127700\pi$
$113$	19.5722	1.84120	0.920600	+	0.390508i	$0.127700\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.0526	1.11426
$118$	0	0
$119$	1.33298	0.122194
$120$	0	0
$121$	−9.80130	−0.891028
$122$	0	0
$123$	20.7898	1.87456
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.00946	0.444517	0.222259	−	0.974988i	$-0.428657\pi$
$127$	5.00946	0.444517	0.222259	+	0.974988i	$0.428657\pi$
$128$	0	0
$129$	17.0764	1.50349
$130$	0	0
$131$	−2.63893	−0.230564	−0.115282	−	0.993333i	$-0.536777\pi$
$131$	−2.63893	−0.230564	−0.115282	+	0.993333i	$0.536777\pi$
$132$	0	0
$133$	−3.67387	−0.318565
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.27064	−0.279430	−0.139715	−	0.990192i	$-0.544619\pi$
$137$	−3.27064	−0.279430	−0.139715	+	0.990192i	$0.544619\pi$
$138$	0	0
$139$	10.3901	0.881276	0.440638	−	0.897685i	$-0.354752\pi$
$139$	10.3901	0.881276	0.440638	+	0.897685i	$0.354752\pi$
$140$	0	0
$141$	1.50234	0.126519
$142$	0	0
$143$	−3.47158	−0.290308
$144$	0	0
$145$	0	0
$146$	0	0
$147$	13.6214	1.12348
$148$	0	0
$149$	−11.2566	−0.922179	−0.461090	−	0.887354i	$-0.652541\pi$
$149$	−11.2566	−0.922179	−0.461090	+	0.887354i	$0.652541\pi$
$150$	0	0
$151$	−7.13736	−0.580831	−0.290415	−	0.956901i	$-0.593793\pi$
$151$	−7.13736	−0.580831	−0.290415	+	0.956901i	$0.593793\pi$
$152$	0	0
$153$	3.80107	0.307299
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.30595	−0.662887	−0.331443	−	0.943475i	$-0.607536\pi$
$157$	−8.30595	−0.662887	−0.331443	+	0.943475i	$0.607536\pi$
$158$	0	0
$159$	−20.4615	−1.62270
$160$	0	0
$161$	−4.76022	−0.375158
$162$	0	0
$163$	−11.6059	−0.909042	−0.454521	−	0.890736i	$-0.650189\pi$
$163$	−11.6059	−0.909042	−0.454521	+	0.890736i	$0.650189\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.6832	1.52313	0.761566	−	0.648087i	$-0.224431\pi$
$167$	19.6832	1.52313	0.761566	+	0.648087i	$0.224431\pi$
$168$	0	0
$169$	−2.94583	−0.226602
$170$	0	0
$171$	−10.4763	−0.801140
$172$	0	0
$173$	19.8536	1.50944	0.754722	−	0.656045i	$-0.227772\pi$
$173$	19.8536	1.50944	0.754722	+	0.656045i	$0.227772\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−25.4628	−1.91390
$178$	0	0
$179$	6.71142	0.501635	0.250817	−	0.968034i	$-0.419301\pi$
$179$	6.71142	0.501635	0.250817	+	0.968034i	$0.419301\pi$
$180$	0	0
$181$	1.49564	0.111170	0.0555852	−	0.998454i	$-0.482298\pi$
$181$	1.49564	0.111170	0.0555852	+	0.998454i	$0.482298\pi$
$182$	0	0
$183$	13.6021	1.00550
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.09485	−0.0800633
$188$	0	0
$189$	−2.78474	−0.202560
$190$	0	0
$191$	15.9320	1.15280	0.576401	−	0.817167i	$-0.304457\pi$
$191$	15.9320	1.15280	0.576401	+	0.817167i	$0.304457\pi$
$192$	0	0
$193$	−17.4673	−1.25732	−0.628661	−	0.777680i	$-0.716397\pi$
$193$	−17.4673	−1.25732	−0.628661	+	0.777680i	$0.716397\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.0436	−1.00057	−0.500283	−	0.865862i	$-0.666771\pi$
$197$	−14.0436	−1.00057	−0.500283	+	0.865862i	$0.666771\pi$
$198$	0	0
$199$	4.58571	0.325072	0.162536	−	0.986703i	$-0.448033\pi$
$199$	4.58571	0.325072	0.162536	+	0.986703i	$0.448033\pi$
$200$	0	0
$201$	−6.90993	−0.487389
$202$	0	0
$203$	−0.240013	−0.0168456
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−13.5741	−0.943462
$208$	0	0
$209$	3.01755	0.208728
$210$	0	0
$211$	−7.23345	−0.497971	−0.248986	−	0.968507i	$-0.580097\pi$
$211$	−7.23345	−0.497971	−0.248986	+	0.968507i	$0.580097\pi$
$212$	0	0
$213$	34.9754	2.39647
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.08776	−0.0738422
$218$	0	0
$219$	32.1550	2.17283
$220$	0	0
$221$	3.17083	0.213293
$222$	0	0
$223$	22.2566	1.49041	0.745207	−	0.666833i	$-0.232351\pi$
$223$	22.2566	1.49041	0.745207	+	0.666833i	$0.232351\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.3541	−0.886342	−0.443171	−	0.896437i	$-0.646147\pi$
$227$	−13.3541	−0.886342	−0.443171	+	0.896437i	$0.646147\pi$
$228$	0	0
$229$	−20.4647	−1.35235	−0.676174	−	0.736742i	$-0.736363\pi$
$229$	−20.4647	−1.35235	−0.676174	+	0.736742i	$0.736363\pi$
$230$	0	0
$231$	3.80598	0.250415
$232$	0	0
$233$	−16.1239	−1.05631	−0.528155	−	0.849148i	$-0.677116\pi$
$233$	−16.1239	−1.05631	−0.528155	+	0.849148i	$0.677116\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.77292	0.634820
$238$	0	0
$239$	4.38637	0.283731	0.141865	−	0.989886i	$-0.454690\pi$
$239$	4.38637	0.283731	0.141865	+	0.989886i	$0.454690\pi$
$240$	0	0
$241$	−14.8343	−0.955558	−0.477779	−	0.878480i	$-0.658558\pi$
$241$	−14.8343	−0.955558	−0.477779	+	0.878480i	$0.658558\pi$
$242$	0	0
$243$	21.7974	1.39831
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.73923	−0.556064
$248$	0	0
$249$	−12.1683	−0.771134
$250$	0	0
$251$	−14.5998	−0.921534	−0.460767	−	0.887521i	$-0.652426\pi$
$251$	−14.5998	−0.921534	−0.460767	+	0.887521i	$0.652426\pi$
$252$	0	0
$253$	3.90983	0.245809
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.34722	0.333550	0.166775	−	0.985995i	$-0.446665\pi$
$257$	5.34722	0.333550	0.166775	+	0.985995i	$0.446665\pi$
$258$	0	0
$259$	11.2612	0.699739
$260$	0	0
$261$	−0.684413	−0.0423641
$262$	0	0
$263$	−18.3754	−1.13307	−0.566537	−	0.824037i	$-0.691717\pi$
$263$	−18.3754	−1.13307	−0.566537	+	0.824037i	$0.691717\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.84832	−0.480310
$268$	0	0
$269$	−15.0812	−0.919516	−0.459758	−	0.888044i	$-0.652064\pi$
$269$	−15.0812	−0.919516	−0.459758	+	0.888044i	$0.652064\pi$
$270$	0	0
$271$	−20.6031	−1.25155	−0.625774	−	0.780004i	$-0.715217\pi$
$271$	−20.6031	−1.25155	−0.625774	+	0.780004i	$0.715217\pi$
$272$	0	0
$273$	−11.0226	−0.667120
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.93417	−0.236381	−0.118191	−	0.992991i	$-0.537709\pi$
$277$	−3.93417	−0.236381	−0.118191	+	0.992991i	$0.537709\pi$
$278$	0	0
$279$	−3.10183	−0.185701
$280$	0	0
$281$	24.7878	1.47872	0.739358	−	0.673312i	$-0.235129\pi$
$281$	24.7878	1.47872	0.739358	+	0.673312i	$0.235129\pi$
$282$	0	0
$283$	−10.2892	−0.611631	−0.305815	−	0.952091i	$-0.598929\pi$
$283$	−10.2892	−0.611631	−0.305815	+	0.952091i	$0.598929\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.6264	−0.627256
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	31.9938	1.87551
$292$	0	0
$293$	−4.20448	−0.245628	−0.122814	−	0.992430i	$-0.539192\pi$
$293$	−4.20448	−0.245628	−0.122814	+	0.992430i	$0.539192\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.28726	0.132720
$298$	0	0
$299$	−11.3234	−0.654848
$300$	0	0
$301$	−8.72832	−0.503092
$302$	0	0
$303$	−7.77446	−0.446631
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.2969	−1.15841	−0.579204	−	0.815183i	$-0.696637\pi$
$307$	−20.2969	−1.15841	−0.579204	+	0.815183i	$0.696637\pi$
$308$	0	0
$309$	−36.1036	−2.05386
$310$	0	0
$311$	−8.03850	−0.455822	−0.227911	−	0.973682i	$-0.573189\pi$
$311$	−8.03850	−0.455822	−0.227911	+	0.973682i	$0.573189\pi$
$312$	0	0
$313$	−29.6525	−1.67606	−0.838028	−	0.545627i	$-0.816292\pi$
$313$	−29.6525	−1.67606	−0.838028	+	0.545627i	$0.816292\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.06410	0.172097	0.0860484	−	0.996291i	$-0.472576\pi$
$317$	3.06410	0.172097	0.0860484	+	0.996291i	$0.472576\pi$
$318$	0	0
$319$	0.197136	0.0110375
$320$	0	0
$321$	−20.5268	−1.14569
$322$	0	0
$323$	−2.75613	−0.153355
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−35.6599	−1.97200
$328$	0	0
$329$	−0.767895	−0.0423354
$330$	0	0
$331$	−11.6185	−0.638609	−0.319305	−	0.947652i	$-0.603449\pi$
$331$	−11.6185	−0.638609	−0.319305	+	0.947652i	$0.603449\pi$
$332$	0	0
$333$	32.1121	1.75973
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.19732	−0.392063	−0.196032	−	0.980598i	$-0.562805\pi$
$337$	−7.19732	−0.392063	−0.196032	+	0.980598i	$0.562805\pi$
$338$	0	0
$339$	−51.0421	−2.77223
$340$	0	0
$341$	0.893440	0.0483825
$342$	0	0
$343$	−16.2932	−0.879752
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.49365	−0.455963	−0.227982	−	0.973665i	$-0.573213\pi$
$347$	−8.49365	−0.455963	−0.227982	+	0.973665i	$0.573213\pi$
$348$	0	0
$349$	−0.281605	−0.0150739	−0.00753697	−	0.999972i	$-0.502399\pi$
$349$	−0.281605	−0.0150739	−0.00753697	+	0.999972i	$0.502399\pi$
$350$	0	0
$351$	−6.62422	−0.353575
$352$	0	0
$353$	7.56460	0.402623	0.201311	−	0.979527i	$-0.435480\pi$
$353$	7.56460	0.402623	0.201311	+	0.979527i	$0.435480\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.47626	−0.183983
$358$	0	0
$359$	−4.24214	−0.223891	−0.111946	−	0.993714i	$-0.535708\pi$
$359$	−4.24214	−0.223891	−0.111946	+	0.993714i	$0.535708\pi$
$360$	0	0
$361$	−11.4037	−0.600197
$362$	0	0
$363$	25.5607	1.34159
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.9623	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$367$	27.9623	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$368$	0	0
$369$	−30.3018	−1.57745
$370$	0	0
$371$	10.4586	0.542982
$372$	0	0
$373$	−9.39715	−0.486566	−0.243283	−	0.969955i	$-0.578224\pi$
$373$	−9.39715	−0.486566	−0.243283	+	0.969955i	$0.578224\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.570933	−0.0294045
$378$	0	0
$379$	−23.9952	−1.23255	−0.616275	−	0.787531i	$-0.711359\pi$
$379$	−23.9952	−1.23255	−0.616275	+	0.787531i	$0.711359\pi$
$380$	0	0
$381$	−13.0641	−0.669294
$382$	0	0
$383$	−1.97888	−0.101116	−0.0505581	−	0.998721i	$-0.516100\pi$
$383$	−1.97888	−0.101116	−0.0505581	+	0.998721i	$0.516100\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.8894	−1.26520
$388$	0	0
$389$	−26.5739	−1.34735	−0.673674	−	0.739028i	$-0.735285\pi$
$389$	−26.5739	−1.34735	−0.673674	+	0.739028i	$0.735285\pi$
$390$	0	0
$391$	−3.57111	−0.180599
$392$	0	0
$393$	6.88202	0.347152
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.32026	0.317205	0.158602	−	0.987343i	$-0.449301\pi$
$397$	6.32026	0.317205	0.158602	+	0.987343i	$0.449301\pi$
$398$	0	0
$399$	9.58103	0.479651
$400$	0	0
$401$	−14.9995	−0.749041	−0.374520	−	0.927219i	$-0.622193\pi$
$401$	−14.9995	−0.749041	−0.374520	+	0.927219i	$0.622193\pi$
$402$	0	0
$403$	−2.58752	−0.128894
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.24947	−0.458479
$408$	0	0
$409$	18.0132	0.890697	0.445348	−	0.895357i	$-0.353080\pi$
$409$	18.0132	0.890697	0.445348	+	0.895357i	$0.353080\pi$
$410$	0	0
$411$	8.52947	0.420728
$412$	0	0
$413$	13.0149	0.640421
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−27.0962	−1.32691
$418$	0	0
$419$	11.2294	0.548594	0.274297	−	0.961645i	$-0.411555\pi$
$419$	11.2294	0.548594	0.274297	+	0.961645i	$0.411555\pi$
$420$	0	0
$421$	37.8927	1.84678	0.923389	−	0.383866i	$-0.125408\pi$
$421$	37.8927	1.84678	0.923389	+	0.383866i	$0.125408\pi$
$422$	0	0
$423$	−2.18970	−0.106467
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.95252	−0.336456
$428$	0	0
$429$	9.05350	0.437107
$430$	0	0
$431$	−14.7151	−0.708803	−0.354402	−	0.935093i	$-0.615315\pi$
$431$	−14.7151	−0.708803	−0.354402	+	0.935093i	$0.615315\pi$
$432$	0	0
$433$	−37.5179	−1.80299	−0.901497	−	0.432786i	$-0.857531\pi$
$433$	−37.5179	−1.80299	−0.901497	+	0.432786i	$0.857531\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.84245	0.470828
$438$	0	0
$439$	−34.8342	−1.66255	−0.831273	−	0.555864i	$-0.812388\pi$
$439$	−34.8342	−1.66255	−0.831273	+	0.555864i	$0.812388\pi$
$440$	0	0
$441$	−19.8536	−0.945411
$442$	0	0
$443$	−9.84325	−0.467667	−0.233833	−	0.972277i	$-0.575127\pi$
$443$	−9.84325	−0.467667	−0.233833	+	0.972277i	$0.575127\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	29.3560	1.38849
$448$	0	0
$449$	27.7439	1.30932	0.654658	−	0.755925i	$-0.272813\pi$
$449$	27.7439	1.30932	0.654658	+	0.755925i	$0.272813\pi$
$450$	0	0
$451$	8.72804	0.410987
$452$	0	0
$453$	18.6134	0.874536
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0953	1.31424	0.657120	−	0.753786i	$-0.271774\pi$
$457$	28.0953	1.31424	0.657120	+	0.753786i	$0.271774\pi$
$458$	0	0
$459$	−2.08911	−0.0975114
$460$	0	0
$461$	5.20430	0.242388	0.121194	−	0.992629i	$-0.461328\pi$
$461$	5.20430	0.242388	0.121194	+	0.992629i	$0.461328\pi$
$462$	0	0
$463$	1.28264	0.0596092	0.0298046	−	0.999556i	$-0.490511\pi$
$463$	1.28264	0.0596092	0.0298046	+	0.999556i	$0.490511\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.5006	−1.13375	−0.566876	−	0.823803i	$-0.691848\pi$
$467$	−24.5006	−1.13375	−0.566876	+	0.823803i	$0.691848\pi$
$468$	0	0
$469$	3.53190	0.163088
$470$	0	0
$471$	21.6610	0.998085
$472$	0	0
$473$	7.16905	0.329633
$474$	0	0
$475$	0	0
$476$	0	0
$477$	29.8233	1.36551
$478$	0	0
$479$	31.1171	1.42178	0.710888	−	0.703306i	$-0.248293\pi$
$479$	31.1171	1.42178	0.710888	+	0.703306i	$0.248293\pi$
$480$	0	0
$481$	26.7877	1.22141
$482$	0	0
$483$	12.4141	0.564861
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.8828	−0.900978	−0.450489	−	0.892782i	$-0.648750\pi$
$487$	−19.8828	−0.900978	−0.450489	+	0.892782i	$0.648750\pi$
$488$	0	0
$489$	30.2668	1.36871
$490$	0	0
$491$	−19.8895	−0.897602	−0.448801	−	0.893632i	$-0.648149\pi$
$491$	−19.8895	−0.897602	−0.448801	+	0.893632i	$0.648149\pi$
$492$	0	0
$493$	−0.180058	−0.00810939
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.8771	−0.801898
$498$	0	0
$499$	−4.68299	−0.209639	−0.104820	−	0.994491i	$-0.533427\pi$
$499$	−4.68299	−0.209639	−0.104820	+	0.994491i	$0.533427\pi$
$500$	0	0
$501$	−51.3316	−2.29333
$502$	0	0
$503$	−39.2233	−1.74888	−0.874440	−	0.485134i	$-0.838771\pi$
$503$	−39.2233	−1.74888	−0.874440	+	0.485134i	$0.838771\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.68239	0.341187
$508$	0	0
$509$	16.9770	0.752494	0.376247	−	0.926519i	$-0.377214\pi$
$509$	16.9770	0.752494	0.376247	+	0.926519i	$0.377214\pi$
$510$	0	0
$511$	−16.4355	−0.727064
$512$	0	0
$513$	5.75786	0.254216
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.630714	0.0277388
$518$	0	0
$519$	−51.7760	−2.27272
$520$	0	0
$521$	−24.6927	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$521$	−24.6927	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$522$	0	0
$523$	41.2117	1.80206	0.901032	−	0.433753i	$-0.142811\pi$
$523$	41.2117	1.80206	0.901032	+	0.433753i	$0.142811\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.816039	−0.0355472
$528$	0	0
$529$	−10.2472	−0.445529
$530$	0	0
$531$	37.1128	1.61056
$532$	0	0
$533$	−25.2776	−1.09489
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17.5026	−0.755294
$538$	0	0
$539$	5.71858	0.246317
$540$	0	0
$541$	−3.70168	−0.159147	−0.0795737	−	0.996829i	$-0.525356\pi$
$541$	−3.70168	−0.159147	−0.0795737	+	0.996829i	$0.525356\pi$
$542$	0	0
$543$	−3.90047	−0.167385
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.0162	−1.32616	−0.663079	−	0.748550i	$-0.730750\pi$
$547$	−31.0162	−1.32616	−0.663079	+	0.748550i	$0.730750\pi$
$548$	0	0
$549$	−19.8255	−0.846134
$550$	0	0
$551$	0.496263	0.0211415
$552$	0	0
$553$	−4.99527	−0.212421
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.6170	−1.72100	−0.860498	−	0.509453i	$-0.829848\pi$
$557$	−40.6170	−1.72100	−0.860498	+	0.509453i	$0.829848\pi$
$558$	0	0
$559$	−20.7625	−0.878162
$560$	0	0
$561$	2.85524	0.120548
$562$	0	0
$563$	6.63255	0.279529	0.139764	−	0.990185i	$-0.455366\pi$
$563$	6.63255	0.279529	0.139764	+	0.990185i	$0.455366\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.93797	−0.333363
$568$	0	0
$569$	−20.9921	−0.880035	−0.440017	−	0.897989i	$-0.645028\pi$
$569$	−20.9921	−0.880035	−0.440017	+	0.897989i	$0.645028\pi$
$570$	0	0
$571$	−9.22867	−0.386208	−0.193104	−	0.981178i	$-0.561855\pi$
$571$	−9.22867	−0.386208	−0.193104	+	0.981178i	$0.561855\pi$
$572$	0	0
$573$	−41.5490	−1.73573
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.17031	−0.0487208	−0.0243604	−	0.999703i	$-0.507755\pi$
$577$	−1.17031	−0.0487208	−0.0243604	+	0.999703i	$0.507755\pi$
$578$	0	0
$579$	45.5526	1.89310
$580$	0	0
$581$	6.21963	0.258034
$582$	0	0
$583$	−8.59021	−0.355770
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.1314	−1.32620	−0.663102	−	0.748529i	$-0.730761\pi$
$587$	−32.1314	−1.32620	−0.663102	+	0.748529i	$0.730761\pi$
$588$	0	0
$589$	2.24911	0.0926730
$590$	0	0
$591$	36.6242	1.50652
$592$	0	0
$593$	−24.9096	−1.02291	−0.511456	−	0.859309i	$-0.670894\pi$
$593$	−24.9096	−1.02291	−0.511456	+	0.859309i	$0.670894\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.9590	−0.489450
$598$	0	0
$599$	30.4971	1.24608	0.623039	−	0.782191i	$-0.285898\pi$
$599$	30.4971	1.24608	0.623039	+	0.782191i	$0.285898\pi$
$600$	0	0
$601$	−4.43000	−0.180703	−0.0903517	−	0.995910i	$-0.528799\pi$
$601$	−4.43000	−0.180703	−0.0903517	+	0.995910i	$0.528799\pi$
$602$	0	0
$603$	10.0714	0.410140
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.4929	−1.15649	−0.578246	−	0.815862i	$-0.696263\pi$
$607$	−28.4929	−1.15649	−0.578246	+	0.815862i	$0.696263\pi$
$608$	0	0
$609$	0.625927	0.0253639
$610$	0	0
$611$	−1.82663	−0.0738976
$612$	0	0
$613$	−5.63970	−0.227785	−0.113893	−	0.993493i	$-0.536332\pi$
$613$	−5.63970	−0.227785	−0.113893	+	0.993493i	$0.536332\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.49860	0.221365	0.110683	−	0.993856i	$-0.464696\pi$
$617$	5.49860	0.221365	0.110683	+	0.993856i	$0.464696\pi$
$618$	0	0
$619$	20.7770	0.835096	0.417548	−	0.908655i	$-0.362889\pi$
$619$	20.7770	0.835096	0.417548	+	0.908655i	$0.362889\pi$
$620$	0	0
$621$	7.46045	0.299377
$622$	0	0
$623$	4.01154	0.160719
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−7.86943	−0.314275
$628$	0	0
$629$	8.44817	0.336850
$630$	0	0
$631$	9.55265	0.380285	0.190142	−	0.981757i	$-0.439105\pi$
$631$	9.55265	0.380285	0.190142	+	0.981757i	$0.439105\pi$
$632$	0	0
$633$	18.8640	0.749778
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−16.5618	−0.656201
$638$	0	0
$639$	−50.9777	−2.01665
$640$	0	0
$641$	−38.2638	−1.51133	−0.755664	−	0.654959i	$-0.772686\pi$
$641$	−38.2638	−1.51133	−0.755664	+	0.654959i	$0.772686\pi$
$642$	0	0
$643$	−2.28036	−0.0899286	−0.0449643	−	0.998989i	$-0.514317\pi$
$643$	−2.28036	−0.0899286	−0.0449643	+	0.998989i	$0.514317\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.3848	−0.604840	−0.302420	−	0.953175i	$-0.597794\pi$
$647$	−15.3848	−0.604840	−0.302420	+	0.953175i	$0.597794\pi$
$648$	0	0
$649$	−10.6898	−0.419613
$650$	0	0
$651$	2.83677	0.111182
$652$	0	0
$653$	24.7373	0.968046	0.484023	−	0.875055i	$-0.339175\pi$
$653$	24.7373	0.968046	0.484023	+	0.875055i	$0.339175\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−46.8669	−1.82845
$658$	0	0
$659$	17.3744	0.676812	0.338406	−	0.941000i	$-0.390112\pi$
$659$	17.3744	0.676812	0.338406	+	0.941000i	$0.390112\pi$
$660$	0	0
$661$	4.18538	0.162793	0.0813963	−	0.996682i	$-0.474062\pi$
$661$	4.18538	0.162793	0.0813963	+	0.996682i	$0.474062\pi$
$662$	0	0
$663$	−8.26917	−0.321148
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.643006	0.0248973
$668$	0	0
$669$	−58.0428	−2.24406
$670$	0	0
$671$	5.71049	0.220451
$672$	0	0
$673$	16.8948	0.651246	0.325623	−	0.945500i	$-0.394426\pi$
$673$	16.8948	0.651246	0.325623	+	0.945500i	$0.394426\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.13351	0.235730	0.117865	−	0.993030i	$-0.462395\pi$
$677$	6.13351	0.235730	0.117865	+	0.993030i	$0.462395\pi$
$678$	0	0
$679$	−16.3531	−0.627576
$680$	0	0
$681$	34.8260	1.33453
$682$	0	0
$683$	26.7588	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$683$	26.7588	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	53.3697	2.03618
$688$	0	0
$689$	24.8784	0.947791
$690$	0	0
$691$	−23.1367	−0.880160	−0.440080	−	0.897959i	$-0.645050\pi$
$691$	−23.1367	−0.880160	−0.440080	+	0.897959i	$0.645050\pi$
$692$	0	0
$693$	−5.54733	−0.210726
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.97191	−0.301957
$698$	0	0
$699$	42.0492	1.59045
$700$	0	0
$701$	41.0017	1.54861	0.774307	−	0.632810i	$-0.218098\pi$
$701$	41.0017	1.54861	0.774307	+	0.632810i	$0.218098\pi$
$702$	0	0
$703$	−23.2843	−0.878182
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.97379	0.149450
$708$	0	0
$709$	−10.7096	−0.402207	−0.201103	−	0.979570i	$-0.564453\pi$
$709$	−10.7096	−0.402207	−0.201103	+	0.979570i	$0.564453\pi$
$710$	0	0
$711$	−14.2443	−0.534204
$712$	0	0
$713$	2.91417	0.109136
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.4392	−0.427204
$718$	0	0
$719$	6.17046	0.230119	0.115060	−	0.993359i	$-0.463294\pi$
$719$	6.17046	0.230119	0.115060	+	0.993359i	$0.463294\pi$
$720$	0	0
$721$	18.4538	0.687254
$722$	0	0
$723$	38.6861	1.43875
$724$	0	0
$725$	0	0
$726$	0	0
$727$	38.9277	1.44375	0.721875	−	0.692024i	$-0.243281\pi$
$727$	38.9277	1.44375	0.721875	+	0.692024i	$0.243281\pi$
$728$	0	0
$729$	−38.9801	−1.44371
$730$	0	0
$731$	−6.54798	−0.242186
$732$	0	0
$733$	31.8680	1.17707	0.588535	−	0.808472i	$-0.299705\pi$
$733$	31.8680	1.17707	0.588535	+	0.808472i	$0.299705\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.90094	−0.106858
$738$	0	0
$739$	−29.7808	−1.09551	−0.547753	−	0.836640i	$-0.684517\pi$
$739$	−29.7808	−1.09551	−0.547753	+	0.836640i	$0.684517\pi$
$740$	0	0
$741$	22.7909	0.837245
$742$	0	0
$743$	−33.4684	−1.22784	−0.613918	−	0.789370i	$-0.710408\pi$
$743$	−33.4684	−1.22784	−0.613918	+	0.789370i	$0.710408\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17.7357	0.648914
$748$	0	0
$749$	10.4919	0.383367
$750$	0	0
$751$	−15.9989	−0.583808	−0.291904	−	0.956448i	$-0.594289\pi$
$751$	−15.9989	−0.583808	−0.291904	+	0.956448i	$0.594289\pi$
$752$	0	0
$753$	38.0747	1.38752
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.6978	−0.752275	−0.376138	−	0.926564i	$-0.622748\pi$
$757$	−20.6978	−0.752275	−0.376138	+	0.926564i	$0.622748\pi$
$758$	0	0
$759$	−10.1964	−0.370105
$760$	0	0
$761$	−22.4458	−0.813660	−0.406830	−	0.913504i	$-0.633366\pi$
$761$	−22.4458	−0.813660	−0.406830	+	0.913504i	$0.633366\pi$
$762$	0	0
$763$	18.2270	0.659861
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.9592	1.11787
$768$	0	0
$769$	30.6260	1.10440	0.552202	−	0.833711i	$-0.313788\pi$
$769$	30.6260	1.10440	0.552202	+	0.833711i	$0.313788\pi$
$770$	0	0
$771$	−13.9449	−0.502215
$772$	0	0
$773$	−10.9578	−0.394126	−0.197063	−	0.980391i	$-0.563140\pi$
$773$	−10.9578	−0.394126	−0.197063	+	0.980391i	$0.563140\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−29.3680	−1.05357
$778$	0	0
$779$	21.9716	0.787215
$780$	0	0
$781$	14.6835	0.525415
$782$	0	0
$783$	0.376161	0.0134429
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.1539	−1.11052	−0.555259	−	0.831677i	$-0.687381\pi$
$787$	−31.1539	−1.11052	−0.555259	+	0.831677i	$0.687381\pi$
$788$	0	0
$789$	47.9209	1.70603
$790$	0	0
$791$	26.0894	0.927631
$792$	0	0
$793$	−16.5383	−0.587294
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.2518	0.540246	0.270123	−	0.962826i	$-0.412936\pi$
$797$	15.2518	0.540246	0.270123	+	0.962826i	$0.412936\pi$
$798$	0	0
$799$	−0.576074	−0.0203800
$800$	0	0
$801$	11.4392	0.404183
$802$	0	0
$803$	13.4994	0.476383
$804$	0	0
$805$	0	0
$806$	0	0
$807$	39.3300	1.38448
$808$	0	0
$809$	−2.66539	−0.0937100	−0.0468550	−	0.998902i	$-0.514920\pi$
$809$	−2.66539	−0.0937100	−0.0468550	+	0.998902i	$0.514920\pi$
$810$	0	0
$811$	−5.54482	−0.194705	−0.0973525	−	0.995250i	$-0.531037\pi$
$811$	−5.54482	−0.194705	−0.0973525	+	0.995250i	$0.531037\pi$
$812$	0	0
$813$	53.7305	1.88441
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.0471	0.631388
$818$	0	0
$819$	16.0658	0.561385
$820$	0	0
$821$	31.2022	1.08896	0.544482	−	0.838773i	$-0.316726\pi$
$821$	31.2022	1.08896	0.544482	+	0.838773i	$0.316726\pi$
$822$	0	0
$823$	−50.4682	−1.75921	−0.879606	−	0.475703i	$-0.842194\pi$
$823$	−50.4682	−1.75921	−0.879606	+	0.475703i	$0.842194\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.7704	0.896123	0.448062	−	0.894003i	$-0.352114\pi$
$827$	25.7704	0.896123	0.448062	+	0.894003i	$0.352114\pi$
$828$	0	0
$829$	4.34228	0.150814	0.0754068	−	0.997153i	$-0.475974\pi$
$829$	4.34228	0.150814	0.0754068	+	0.997153i	$0.475974\pi$
$830$	0	0
$831$	10.2599	0.355911
$832$	0	0
$833$	−5.22317	−0.180972
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.70480	0.0589264
$838$	0	0
$839$	−44.9482	−1.55178	−0.775892	−	0.630866i	$-0.782700\pi$
$839$	−44.9482	−1.55178	−0.775892	+	0.630866i	$0.782700\pi$
$840$	0	0
$841$	−28.9676	−0.998882
$842$	0	0
$843$	−64.6438	−2.22645
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−13.0649	−0.448917
$848$	0	0
$849$	26.8331	0.920910
$850$	0	0
$851$	−30.1693	−1.03419
$852$	0	0
$853$	−15.7009	−0.537590	−0.268795	−	0.963197i	$-0.586625\pi$
$853$	−15.7009	−0.537590	−0.268795	+	0.963197i	$0.586625\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.77401	0.231396	0.115698	−	0.993284i	$-0.463090\pi$
$857$	6.77401	0.231396	0.115698	+	0.993284i	$0.463090\pi$
$858$	0	0
$859$	−36.0071	−1.22855	−0.614273	−	0.789094i	$-0.710551\pi$
$859$	−36.0071	−1.22855	−0.614273	+	0.789094i	$0.710551\pi$
$860$	0	0
$861$	27.7124	0.944437
$862$	0	0
$863$	35.5716	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$863$	35.5716	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.60789	−0.0885685
$868$	0	0
$869$	4.10289	0.139181
$870$	0	0
$871$	8.40152	0.284675
$872$	0	0
$873$	−46.6320	−1.57825
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24.1451	−0.815321	−0.407660	−	0.913134i	$-0.633655\pi$
$877$	−24.1451	−0.815321	−0.407660	+	0.913134i	$0.633655\pi$
$878$	0	0
$879$	10.9648	0.369834
$880$	0	0
$881$	32.9124	1.10885	0.554423	−	0.832235i	$-0.312939\pi$
$881$	32.9124	1.10885	0.554423	+	0.832235i	$0.312939\pi$
$882$	0	0
$883$	−15.5885	−0.524594	−0.262297	−	0.964987i	$-0.584480\pi$
$883$	−15.5885	−0.524594	−0.262297	+	0.964987i	$0.584480\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.59951	−0.288743	−0.144372	−	0.989524i	$-0.546116\pi$
$887$	−8.59951	−0.288743	−0.144372	+	0.989524i	$0.546116\pi$
$888$	0	0
$889$	6.67750	0.223956
$890$	0	0
$891$	6.51989	0.218425
$892$	0	0
$893$	1.58773	0.0531315
$894$	0	0
$895$	0	0
$896$	0	0
$897$	29.5301	0.985982
$898$	0	0
$899$	0.146934	0.00490053
$900$	0	0
$901$	7.84602	0.261389
$902$	0	0
$903$	22.7625	0.757488
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.9819	1.49360	0.746800	−	0.665049i	$-0.231589\pi$
$907$	44.9819	1.49360	0.746800	+	0.665049i	$0.231589\pi$
$908$	0	0
$909$	11.3315	0.375842
$910$	0	0
$911$	−50.9800	−1.68904	−0.844522	−	0.535521i	$-0.820115\pi$
$911$	−50.9800	−1.68904	−0.844522	+	0.535521i	$0.820115\pi$
$912$	0	0
$913$	−5.10852	−0.169067
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.51763	−0.116163
$918$	0	0
$919$	13.0346	0.429973	0.214986	−	0.976617i	$-0.431029\pi$
$919$	13.0346	0.429973	0.214986	+	0.976617i	$0.431029\pi$
$920$	0	0
$921$	52.9321	1.74417
$922$	0	0
$923$	−42.5252	−1.39974
$924$	0	0
$925$	0	0
$926$	0	0
$927$	52.6221	1.72834
$928$	0	0
$929$	37.8748	1.24263	0.621316	−	0.783560i	$-0.286598\pi$
$929$	37.8748	1.24263	0.621316	+	0.783560i	$0.286598\pi$
$930$	0	0
$931$	14.3957	0.471801
$932$	0	0
$933$	20.9635	0.686314
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.9509	0.749772	0.374886	−	0.927071i	$-0.377682\pi$
$937$	22.9509	0.749772	0.374886	+	0.927071i	$0.377682\pi$
$938$	0	0
$939$	77.3303	2.52358
$940$	0	0
$941$	−33.8175	−1.10242	−0.551209	−	0.834367i	$-0.685833\pi$
$941$	−33.8175	−1.10242	−0.551209	+	0.834367i	$0.685833\pi$
$942$	0	0
$943$	28.4685	0.927064
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.0614	−0.326953	−0.163476	−	0.986547i	$-0.552271\pi$
$947$	−10.0614	−0.326953	−0.163476	+	0.986547i	$0.552271\pi$
$948$	0	0
$949$	−39.0960	−1.26911
$950$	0	0
$951$	−7.99082	−0.259120
$952$	0	0
$953$	38.8829	1.25954	0.629770	−	0.776781i	$-0.283149\pi$
$953$	38.8829	1.25954	0.629770	+	0.776781i	$0.283149\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.514109	−0.0166188
$958$	0	0
$959$	−4.35970	−0.140782
$960$	0	0
$961$	−30.3341	−0.978519
$962$	0	0
$963$	29.9184	0.964106
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.40033	0.173663	0.0868315	−	0.996223i	$-0.472326\pi$
$967$	5.40033	0.173663	0.0868315	+	0.996223i	$0.472326\pi$
$968$	0	0
$969$	7.18768	0.230902
$970$	0	0
$971$	−10.1469	−0.325629	−0.162815	−	0.986657i	$-0.552057\pi$
$971$	−10.1469	−0.325629	−0.162815	+	0.986657i	$0.552057\pi$
$972$	0	0
$973$	13.8498	0.444004
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.09855	−0.291088	−0.145544	−	0.989352i	$-0.546493\pi$
$977$	−9.09855	−0.291088	−0.145544	+	0.989352i	$0.546493\pi$
$978$	0	0
$979$	−3.29490	−0.105306
$980$	0	0
$981$	51.9754	1.65945
$982$	0	0
$983$	−49.1437	−1.56744	−0.783721	−	0.621113i	$-0.786681\pi$
$983$	−49.1437	−1.56744	−0.783721	+	0.621113i	$0.786681\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.00258	0.0637429
$988$	0	0
$989$	23.3836	0.743554
$990$	0	0
$991$	−43.2065	−1.37250	−0.686249	−	0.727366i	$-0.740744\pi$
$991$	−43.2065	−1.37250	−0.686249	+	0.727366i	$0.740744\pi$
$992$	0	0
$993$	30.2997	0.961531
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.5452	1.53744	0.768721	−	0.639584i	$-0.220893\pi$
$997$	48.5452	1.53744	0.768721	+	0.639584i	$0.220893\pi$
$998$	0	0
$999$	−17.6492	−0.558395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bz.1.1		5
4.3	odd	2		425.2.a.i.1.1	✓	5
5.4	even	2		6800.2.a.cd.1.5		5
12.11	even	2		3825.2.a.bq.1.5		5
20.3	even	4		425.2.b.f.324.10		10
20.7	even	4		425.2.b.f.324.1		10
20.19	odd	2		425.2.a.j.1.5	yes	5
60.59	even	2		3825.2.a.bl.1.1		5
68.67	odd	2		7225.2.a.x.1.1		5
340.339	odd	2		7225.2.a.y.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.1	✓	5	4.3	odd	2
425.2.a.j.1.5	yes	5	20.19	odd	2
425.2.b.f.324.1		10	20.7	even	4
425.2.b.f.324.10		10	20.3	even	4
3825.2.a.bl.1.1		5	60.59	even	2
3825.2.a.bq.1.5		5	12.11	even	2
6800.2.a.bz.1.1		5	1.1	even	1	trivial
6800.2.a.cd.1.5		5	5.4	even	2
7225.2.a.x.1.1		5	68.67	odd	2
7225.2.a.y.1.5		5	340.339	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60789 q^{3} +1.33298 q^{7} +3.80107 q^{9} -1.09485 q^{11} +3.17083 q^{13} +1.00000 q^{17} -2.75613 q^{19} -3.47626 q^{21} -3.57111 q^{23} -2.08911 q^{27} -0.180058 q^{29} -0.816039 q^{31} +2.85524 q^{33} +8.44817 q^{37} -8.26917 q^{39} -7.97191 q^{41} -6.54798 q^{43} -0.576074 q^{47} -5.22317 q^{49} -2.60789 q^{51} +7.84602 q^{53} +7.18768 q^{57} +9.76375 q^{59} -5.21577 q^{61} +5.06675 q^{63} +2.64963 q^{67} +9.31305 q^{69} -13.4114 q^{71} -12.3299 q^{73} -1.45941 q^{77} -3.74745 q^{79} -5.95506 q^{81} +4.66596 q^{83} +0.469570 q^{87} +3.00946 q^{89} +4.22665 q^{91} +2.12814 q^{93} -12.2681 q^{97} -4.16160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - q^{7} + 6 q^{9} - 4 q^{11} - 3 q^{13} + 5 q^{17} - 6 q^{19} - 5 q^{21} - 4 q^{23} + 5 q^{27} + 2 q^{29} - 21 q^{31} + 12 q^{33} - 2 q^{37} - 23 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} + 10 q^{49}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q - q^3 - q^7 + 6 * q^9 - 4 * q^11 - 3 * q^13 + 5 * q^17 - 6 * q^19 - 5 * q^21 - 4 * q^23 + 5 * q^27 + 2 * q^29 - 21 * q^31 + 12 * q^33 - 2 * q^37 - 23 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 + 10 * q^49 - q^51 + 21 * q^53 - 20 * q^57 - 12 * q^59 - 2 * q^61 - 8 * q^63 + 12 * q^67 - 21 * q^71 + 22 * q^73 - 32 * q^77 - 41 * q^79 - 35 * q^81 + 8 * q^83 + 4 * q^87 - 10 * q^89 - 13 * q^91 + 9 * q^93 - 20 * q^97 + 14 * q^99

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